Aircraft Stability
This page was created by Felipe Ortega--Fortega3 14:01, October 15, 2009 (UTC) Aircraft Stability Aircraft stability is the dependency of an aircraft to return to a state of equilibrium after a perturbation. Typically, a coordinate system is attached to the center of gravity of the aircraft in order to describe the dynamics or responses to perturbations. This is done because forces acting on an airplane create moments and rotations naturally about the center of gravity. Also, measurements in flight data are provided by on-body sensors, comparable to what a pilot "feels". The concept of aircraft stability was not born with the aircraft itself. The Wrights' remark:"The balancing of a gliding or flying machine is very simple in theory. It merely consists in causing the center of pressure to coincide with the center of gravity." Consequently, they must have enhanced their difficulties considerably by flying machines which were sometimes unstable (Ref.1). Static Stability Static stability is concerned with the forces and moments produced by a small disturbance from a trim condition. It determines whether or not a body will return, of its own accord, to equilibrium once the perturbation is removed. A body is statically unstable when it tends to diverge from its equilibrium position and it possesses neutral static stability if it remains in the perturbed state (Ref. 2). If a gust causes a change in a nose up or down sense, this is known as a force which causes rotation about the x-axis, and requires analysis of longitudinal static stability. Longitudinal static stability requires an aircraft to return to straight and level flight after a change in alpha or angle of attack is induced by a perturbation. In other words, a pitching moment is induced by a wind gust and if one were to analyze the contributing forces creating moments about the y-axis, the moments should revert the airplane back into a trim condition. These forces are the airplanes weight (which may be idealized as a point mass located at the center of gravity), the aerodynamic forces induced by the wings and empennage, and the responses generated by control surfaces. Figure 2 illustrates the pitching moment coefficient of two different airplanes as they experience a change in angle of attack (aoa). By convention, a positive pitching moment is one which results in a nose-up configuration of the airplane. The black point on the graph depicts the plane flying at a point of equilibrium and the red points show a perturbed state of both airplanes. Notice airplane 1 will continue on a path to a higher aoa given a nose-up perturbation, resulting in an even further divergence from equilibrium. Airplane 2 will pitch back up given its pitching moment coefficient, and return to equilibrium after a nose-down perturbation. Therefore airplane 2 is longitudinally stable. Put simply, longitudinal static stability requires the rate of change of the pitching moment with respect to angle of attack to be negative. Many texts will illustrate the same aspect with respect to the lift coefficient instead. There are several features which effect the pitch moment, both fixed aircraft structures and control surfaces. Wings, horizontal stabilizers and elevators (for the simplest airplane configuration) all contribute to the above equation. About the x-axis, the same principle is referred to as lateral static stability. This involves the rolling motion which results from a side wind gust. The features which effect aircraft lateral stability are again based mostly on the geometric structure of the airplane. Specifically, the location, orientation, and sweep of the wings and the design of the vertical tail. There are several configurations in which wing mountings may be considered and each will uniquely effect the stability of the aircraft. Typically, the effect of the wings and tail must be analyzed simultaneously because the two produce a coupled response. All of these variations lead to trade-offs in aircraft design. Structural features which may go unnoticed to the common spectator may be interpreted by more experienced viewers as design choices which effect the different aspects of flight stability. Similar to the pitch moment coefficient analysis, a roll moment coefficient may be plotted versus beta ( ) or side-slip angle (positive if nose-left from x-axis). Through the same analysis, one can conclude a negative rate of change of the roll moment with respect to beta is laterally stable. Conceptually, an airplane should roll back and return to straight and level flight when a side gust occurs. The final component of static stability is directional stability. This is a resulting balance of forces and moments about the z-axis, also known as the "yaw" of the aircraft. The contributing structures to this analysis are mainly the fuselage and vertical tail of the aircraft. Basically the fuselage is a destabilizing structure in the presence of a side gust, and the tail is a stabilizing structure. This time, with the definition of side-slip, beta, and the yaw moment coefficient as positive for a nose-right moment, the following graph is generated to characterize two different responses. Here it is clear that airplane 1 will return to a point of equilibrium and airplane 2 will amplify the effect of the side gust and become more unstable. For each of the cases presented here, a more complex derivation of the structural components and control surface contributions would be added to the master equations in order to understand how the system, as a whole, meets the requirements of stability. Dynamic Stability Understanding the dynamic stability of an aircraft will require more detail. Apart from the response of an aircraft to small perturbations during straight and level cruise flight, we would also like to understand the response of an aircraft when it undergoes a maneuver about its trim configuration. A series of equations of motion (EOM) are utilized to describe the dynamic response of the aircraft to these maneuvers. The motion usually consists of oscillations about the equilibrium position. A body is dynamically stable if the oscillations eventually damp out (Ref. 2). Looking at the source of these equations, one must be aware that the complete set of information includes a description of forces, moments, position and orientation (attitude). Each of these four sets must be derived in the three directions of the body-centered coordinate system; this alone requires the transformation of coordinate axis from other orientations not described here; but are often more convenient to describe specific forces, etc. Thus the complete dynamic response is modeled by twelve highly coupled, non-linear, differential equations. As a side note, these twelve EOM are only capable of being solved by computational numerical methods. Aeronautic software, such as flight simulator, employ these equations as an extremely accurate model. As is common in many engineering practices, it is convenient to simplify these equations in order to use non-numerical methods to interpret the dynamics of the system. By making the assumptions of small angles and considering maneuvers are first order perturbations about the trim condition, we may reduce the system to twelve equations which naturally decouple. The result is six equations with only longitudinal terms and six with both directional and lateral terms. At this point in the analysis, it is also convenient to recall the operations of linear algebra in regards to solving systems of ordinary differential equations. The purpose of this decomposition is to make more specific assumptions about the particular conditions, and break the dynamic responses into flight modes. As a last assumption, we ignore the instantaneous position of the aircraft at the time of calculation, since its dynamic response does not depend on whether it is flying over point A or B, as well as its yaw angle (for convenience), and the system reduces into common flight modes well understood in aerospace industries. These modifications are quite involved and are the basis of any Aircraft Stability and Control '' course. To re-iterate, the decomposition is as follows: *12 non-linear highly coupled ODE for accurate flight dynamics *12 linear decoupled ODE for trim maneuvering *2 sets of 6 ODE for longitudinal/lateral-directional dynamics *2 sets of 4 ODE by neglecting position/yaw angle *decompose into modes The dynamics of the airplane response can actually be modeled in the same manner as groupings of mass, spring, damper systems. In general, there are four types of solutions. A stable response is illustrated in the bottom cases. Thus the eigenvalues of the characteristic equation must either be real and negative, or complex with a negative real component. The oscillatory nature of the complex solution is either diverging or converging within an exponential envelope (red line). These variations in response lead to different flight mode interpretations. Flight Modes Longitudinal Modes Again, the roots of the characteristic equation dictate the response of the system, and that response will typically take one of four forms. In regards to longitudinal flight modes, there are two forms; the damped short period mode and the lightly damped phugoid mode. The short period mode involves mainly pitching motion with no change in forward speed (Ref. 3) . Neglecting the gravitational terms of the equations, the matrix of ODE decomposes into a direct relationship between pitch rate and change in aoa. This may be interpreted as a pilot command to quickly pitch up or down, then the aircraft response lagging the command, perhaps overshooting and correcting, however dampening this effect very quickly. In other words, this response may be thought of as a phase lag between pitch rate and rate of change of aoa. The lightly damped phugoid mode requires neglecting the components of the equations with respect to changes in aoa and quick changes in pitch (pitch rate). This leads to a correlation between forward velocity and pitch. This response may be conceptually understood as a form of energy conservation. A slight change in forward velocity will lead to increased lift, transferring kinetic energy into potential energy. At the peak of the oscillation, the airplane will descend and consequently speed up again. This oscillation is very slow, hence lightly damped, and if not corrected for, may have a period of several minutes as in commercial aircraft. Lateral-directional Modes Lateral-directional modes for subsonic aircraft include roll, spiral and dutch-roll. The derivation for all flight modes may be rather involved and so only a conceptual overview is provided here. For in depth formulations, see the references listed below. Roll and spiral modes are non-oscillatory responses. An input to the ailerons of an airplane will excite the roll mode. Its motion is almost purely a rolling return to trim with slight yaw and side-slip. A stable response is known as "roll convergence" which is analogous to roll mode. An unstable response is "roll divergence". The spiral mode is more complex and interesting. Consider a wind gust which induces a positive roll moment. This roll angle increase will cause an increase in yaw rate due to the reorientation of aerodynamic forces relative to a constant weight vector, and at the same time the airplane is speeding up as it loses altitude. The result is a descending spiral in which the spirals get tighter and faster. The final mode is the dutch-roll and it is an oscillatory response which couples a rolling, yaw, side-slip motion. The rolling dominates the motion. Basically, the nose of the aircraft will yaw side to side without much change in the direction of flight, then a roll "corrects" the shift with a slight phase lag. The nose of the aircraft essentially traces figure eights in space. This mode is actually quite complex and very dangerous to pilots if they are on a landing approach for example. Final Note Careful analysis of the derivation of stability requirements and modal decomposition leads to an interesting aircraft design dilemna. The structural components which ensure static stability often worsen the response to dynamic flight modes. For example, a large vertical tail lends a directionally stable airplane, however it will worsen the spiral mode of the same airplane. Therefore, careful consideration in the design constraints must increase in scope to include both static and dynamic stability. Also, aircraft with a great deal of stability are often not very maneuverable. There is a definite trade off between the two qualities. This principle is analogous to the driving characteristics of a school bus for example; very stable but not very exciting to drive. The flight modes mentioned are no longer of great concern to aeronautic industries since they are often corrected for by computer control. Their derivations took place decades ago to better understand the physics of flight and designs which lead to a more stable aircraft. References 1. Irving, F.G. ''An Introduction to the Longitudinal Static Stability of Low-speed Aircraft. Chapter 1. Pergamon Press 1966. 2. Dickinson, B. Aircraft Stability and Control for Pilots and Engineers. Page 24. Sir Isaac Pitman and Sons LTD 1968. 3. Babister, A. Aircraft Dynamic Stability and Response. Page 75. Pergamon Press Inc. 1980